The reals however are a different problem, and it's not scientifically possible to prove that the ratio between the length and radius of any object is exactly pi (that it is a perfect circle). However, it's also impossible to prove scientifically that it is 3 or 3.14 or any other number.
Now my use of "unscientific" is more of a hyperbole or click-bait. I thought I explained my actual claim pretty well - that you can't measurably/scientifically distinguish between a universe that contains actual infinities and one that only contains some arbitrarily large numbers.
There's a difference between something not being instantiated in this universe and being unscientific, though.
If we produce a model of the universe that doesn't make a single incorrect prediction given all data available, and it predicts infinities to exist in some strange but quite real cases, is it unscientific?
Of course exactitude exists. For example, two electrons have exactly the same charge. A photon has exactly 0 charge.
> There's a difference between something not being instantiated in this universe and being unscientific, though.
Well, science is a particular way of studying what exists. Studying something that doesn't exist is unscientific (of course, you can use science to try to determine IF something exists).
But there are also things that are outside the reach of the methods of science, so they are unscientific in this sense. Questions such as "did some god create the universe" are unscientific because it is simply impossible to apply the methods of science to arrive at an answer to this question.
Similarly, asking "is the universe infinite in size" is unscientific, because it is impossible to apply the methods of science and arrive at a definite answer to this question.
> If we produce a model of the universe that doesn't make a single incorrect prediction given all data available, and it predicts infinities to exist in some strange but quite real cases, is it unscientific?
If it predicts actual infinities exist in certain conditions, than it is not going to be a testable theory in those conditions. It may still be a perfectly workable model, just as GR is perfectly workable despite predicting singularities at the center of black holes. That doesn't mean that the singularities exist, it means that GR breaks down at certain points.
But even if you had a physical theory that relied on something like a Banach-Tarski construction, you could never distinguish between an actual infinity of points, leading to two perfectly solid, perfectly identical spheres; and an arbitrarily large number of points, leading either to two perfectly solid but slightly different-sized spheres; or two identically-sized spheres with small holes.
Of course, without some need to specify the number of points, you would be well positioned to use the infinite variant. But if someone asked you if this means that the sphere really has an infinite number of points, the answer would have to be that you can't be sure.
Aren't claims like this unscientific according to your standard? You will never be able to measure that two electrons have the same charge to infinite decimal precision. You might have a theory that says they should have the same charge, but you won't be able to test that theory to infinite precision either.
>Studying something that doesn't exist is unscientific
What about things that could exist, might exist, or even aren't expressly forbidden from existing? These have all been used as perfectly valid reasons for scientific inquiry, historically.
Asking "did some god create the universe" is unscientific by your reasoning so long as it is known that there is no in-universe trace or evidence that it was indeed created by a god. Proving that is proving a negative. I think it is not impossible for us to prove that the universe was created by a god, if we found some hidden message in subatomic particles or cosmic dust or something. It does certainly feel impossible that we will prove that the universe wasn't created by a god, though. The inquiry is deemed unscientific because we have no reason to go down that pathway, not because the question is fundamentally intractable.
Multiverse theory, on the other hand, would qualify as unscientific by your reasoning. If it were true, the different universes would be fundamentally inaccessible, according to our understanding. The model does not suggest that evidence could even possibly exist, as far as I understand.
A result being untestable doesn't, in my opinion, lead to it being unscientific. We cannot test whether black holes exist, except by looking for them. We cannot test whether wormholes exist, except by looking for them. These are predictions that we cannot "test" except by looking at the universe and seeing what we find, and even then we are not guaranteed a positive result, just because maybe it is the case that our model is correct but there was never the appropriate state of the universe to prove our prediction.
Of course if something was actually infinite, you wouldn't be able to measure it to be so, but if the model (that you have shown to be correct in other case) predicts an actual infinity and you keep counting more and more orders of magnitude, does it not make sense to assume your model is correct? Is that unscientific? Just like we assume that the charge on electrons is constant despite not actually measuring it always everywhere.
> I think it is not impossible for us to prove that the universe was created by a god, if we found some hidden message in subatomic particles or cosmic dust or something.
That's actually a good point, there could be scientific proof of some intelligent creator in principle. The fact that there is no reason a priori to believe that we will find such a proof is a problem, but I don't think it would be enough to deem the theory unscientific. Otherwise, many actually used theories would be unscientific - for example, there is no scientific reason to expect supersimmetry to exist, but that doesn't make the search for supersimmetry unscientific.
> Multiverse theory, on the other hand, would qualify as unscientific by your reasoning.
Yes, multiverse theory is unscientific by my definition. I don't believe speculation about a multiverse can be considered science in any meaningful sense. Just like simulation theory, it is using science-sounding terminology for idle speculation (though the universe being a simulation could similarly be proven by the same kind of evidence as the intelligent creator idea, to be fair).
> These are predictions that we cannot "test" except by looking at the universe and seeing what we find, and even then we are not guaranteed a positive result
But this is exactly the definition of a test. It's true that you can't prove that something doesn't exist in this way, but saying that something is untestable goes beyond that. An untestable hypothesis is one that by definition doesn't make any predictions about the universe. Multiverse theory is in this bucket - whether you believe it to be true or not, you won't expect to see anything different in the world.
> Of course if something was actually infinite, you wouldn't be able to measure it to be so, but if the model (that you have shown to be correct in other case) predicts an actual infinity and you keep counting more and more orders of magnitude, does it not make sense to assume your model is correct?
Of course it's OK to assume your model is correct, and infinity will likely be the simplest assumption in this case. However, any model that predicts an infinity can be replaced with an equivalent model that makes all the same measurable predictions but replaces the infinity with some arbitrarily large but finite number (or arbitrarily small but not infinitesimal). This second model may well be harder to work with and will contain an extra assumption (an explicit upper bound for the infinite quantity), so I wouldn't advocate for its use. But it would have to be accepted that it is not empirically distinguishable from the infinity based model.
Perhaps the problem here is one of mixing intuition (the idea of 'an object') with rigorous physics and mathematics, perhaps this is where I am going a bit wrong.
